CONDITIONS
FRANC¸ OIS BOLLEY, ARNAUD GUILLIN, AND XINYU WANG
Abstract. Nash and Sobolev inequalities are known to be equivalent to ultracontractive properties of heat-like Markov semigroups, hence to uniform on-diagonal bounds on their kernel densities. In non ultracontractive settings, such bounds can not hold, and (necessarily weaker, non uniform) bounds on the semigroups can be derived by means of weighted Nash (or super-Poincar´e) inequalities. The purpose of this note is to show how to check these weighted Nash inequalities in concrete examples of reversible diffusion Markov semigroups in Rd, in a very simple and general manner. We also deduce off-diagonal bounds for the Markov kernels of the semigroups, refining E. B. Davies’ original argument.
The (Gagliardo-Nirenberg-) Nash inequality inRd states that
kfk1+2/dL2(dx)≤C(d)k∇fkL2(dx)kfk2/dL1(dx) (1) for functionsf onRdand is a powerful tool when studying smoothing properties of parabolic partial differential equations onRd.
In a general way, let (Pt)t≥0 be a symmetric Markov semigroup on a spaceE, with Dirichlet formE and (finite or not) invariant measureµ. Then the Nash inequality
kfk2+4/dL2(dµ) ≤
C1E(f, f) +C2kfk2L2(dµ)
kfk4/dL1(dµ) (2) for a positive parameterd, or more generally
Φ kfk2L2(dµ)
kfk2L1(dµ)
!
≤ E(f, f) kfk2L1(dµ)
(3) for an increasing convex function Φ, is equivalent, up to constants and under adequate hy- potheses on Φ, to the ultracontractivity bound
kPtfkL∞(dµ) ≤n−1(t)kfkL1(dµ), t >0 where
n(t) = Z +∞
t
1 Φ(x)dx;
for instance n−1(t) ≤ Ct−d/2 for 0 < t≤ 1 in the case of (2). We refer in particular to [6]
and [13] in this case, and in the general case of (3) to the seminal work [9] by T. Coulhon where the equivalence was first obtained; see also [3, Chap. 6]. Let us observe that Nash inequalities are adapted to smoothing properties of the semigroup for small times, but can
Date: July 25, 2014.
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also be useful for large times. This is in turn equivalent to uniform bounds on the kernel density ofPt with respect toµ, in the sense that for µ-almost everyx inE one can write
Ptf(x) = Z
E
f(y)pt(x, y)dµ(y) with pt(x, y)≤n−1(t) (4) forµ⊗µ-almost every (x, y) inE×E.Observe finally that (3) is equivalent to its linearised form
kfk2L2(dµ) ≤uE(f, f) +b(u)kfk2L1(dµ), u < u0
for a decreasing positive functionb(u) related to Φ: this form was introduced by F.-Y. Wang [17]
under the name of super-Poincar´e inequality to characterize the generators Lwith empty es- sential spectrum. For certain b(u) it is equivalent to a logarithmic Sobolev inequality for µ, hence, to hypercontractivity only (and not ultracontractivity) of the semigroup.
Moreover, relevant Gaussian off-diagonal bounds on the density pt(x, y) for x6=y, such as pt(x, y)≤C(ε)t−d/2e−d(x,y)2/(4t(1+ε)), t >0
for all ε > 0, have first been obtained by E. B. Davies [12] for the heat semigroup on a Riemannian manifoldE, and by using a family of logarithmic Sobolev inequalities equivalent to (2). Such bounds have been turned optimal in subsequent works, possibly allowing forε= 0 and the optimal numerical constantCwhen starting from the optimal so-called entropy-energy inequality, and extended to more general situations: see for instance [3, Sect. 7.2], [6] and [13]
for a presentation of the strategy based on entropy-energy inequalities, and [5], [10, Sect. 2]
and [11] and the references therein for a presentation of three other ways of deriving off- diagonal bounds from on-diagonal ones (namely based on an integrated maximum principle, finite propagation speed for the wave equation and a complex analysis argument).
In the more general setting where the semigroup is not ultracontractive, then the uniform bound (4) cannot hold, but only (for instance on the diagonal)
pt(x, x)≤n−1(t)V(x)2 (5)
for a nonnegative function V. Such a bound is interesting since it provides information on the semigroup : for instance if V is in L2(µ), then it ensures that Pt is Hilbert-Schmidt, and in particular has a discrete spectrum. It has been shown to be equivalent to a weighted super-Poincar´e inequality
kfk2L2(dµ)≤uE(f, f) +b(u)kf Vk2L1(dµ), u < u0 (6) as in [18], where sharp estimates on high-order eigenvalues are derived, and, as in [2], to a weighted Nash inequality
Φ kfk2L2(dµ)
kf Vk2L1(dµ)
!
≤ E(f, f) kf Vk2L1(dµ)
· (7)
The purpose of this note is twofold. First, to give simple and easy to check sufficient criteria on the generator of the semigroup for the weighted inequalities (6)-(7) to hold : for this, we use Lyapunov conditions, which have revealed an efficient tool to diverse functional inequalities (see [1] or [8] for instance) : we shall see how they allow to recover and extend examples considered in [2] and [18], in a straightforward way (see Example 8). Then, to derive off- diagonal bounds on the kernel density of the semigroup, which will necessarily be non uniform in our non ultracontractive setting. For this we refine Davies’ original ideas of [12]: indeed,
we combine his method with the (weighted) super-Poincar´e inequalities derived in a first step, instead of the families of logarithmic Sobolev inequalities or entropy-energy inequalities used in the ultracontractive cases of [3], [6] and [12]-[13]; we shall see that the method recovers the optimal time dependence when written for (simpler) ultracontractive cases, and give new results in the non ultracontractive case (extending the scope of [2] and [18]). Instead, we could have first derived on-diagonal bounds, such as (5), and then use the general results mentionned above (in particular in [11]) and giving off-diagonal bounds from on-diagonal bounds; but we will see here that, once the inequality (6)-(7) has been derived, the off-diagonal bounds come without further assumptions nor much more effort than the on-diagonal ones.
To make this note as short and focused on the method as possible, we shall only present in detail the situation whereU is aC2 function onRdwith Hessian bounded by below, possibly by a negative constant, and such that R
e−Udx = 1. The differential operator L defined by Lf = ∆f − ∇U · ∇f for C2 functions f on Rd generates a Markov semigroup (Pt)t≥0, defined for all t≥0 by our assumption on the Hessian ofU. It is symmetric inL2(µ) for the invariant measure dµ(x) = e−U(x)dx. We refer to [3, Chap. 3] for a detailed exposition of the background on Markov semigroups. Let us point out that the constants obtained in the statements do not depend on the lower bound of the Hessian ofU, and that the method can be pursued in a more general setting, see Remarks 3, 7 and 11.
We shall only seek upper bounds on the kernel, leaving lower bounds or bounds on the gradient aside (as done in the ultracontractive setting in [10, Sect. 2], [13] or [16] for instance).
Let us finally observe that S. Boutayeb, T. Coulhon and A. Sikora [5, Th. 1.2.1] have most recently devised a general abstract framework, including a functional inequality equivalent to the more general boundpt(x, x)≤m(t, x) than (5), and to the corresponding off-diagonal bound. The derivation of simple practical criteria on the generator ensuring the validity of such (possibly optimal) bounds is an interesting issue, that should be considered elsewhere.
Definition 1. Letξ be aC1 positive increasing function on (0,+∞) and φ be a continuous positive function onRd, with φ(x)→ +∞ as|x| goes to +∞. A C2 map W ≥1 on Rd is a ξ-Lyapunov function with rate φif there existb, r0 ≥0 such that for all x∈Rd
LW
ξ(W)(x)≤ −φ(x) +b1|x|≤r0. We first state our general result:
Proposition 2. In the notation of Definition 1, assume that there exists a ξ-Lyapunov func- tion W with rate φ. Then there exist C and s0 > 0 such that for any positive continuous functionV on Rd
Z
f2dµ≤s Z
|∇f|2(1∨1/ξ′(W))dµ + C g◦ψ4 s
1
s ∨h◦ψ4 s
d/2Z
|f|V dµ2
(8) for all smooth f and s≤s0. Here, for r >0
g(r) = sup
|x|≤r
eU(x)
V(x)2, h(r) = sup
|x|≤r
|∇U(x)|2, ψ(r) = inf{u≥0; inf
|x|≥uφ(x)≥r}.
Here and below a∨b stands for max{a, b}.
Remark 3. Proposition 2 can be extended from theRd case to the case of ad-dimensional connected complete Riemannian manifold M. If M has a boundary ∂M we then have to suppose that ∂nW ≤ 0 for the inward normal vector n to the boundary, namely that the vector ∇W is outcoming at the boundary. We also have to check that a local super-Poincar´e inequality holds: this inequality can easily be obtained in the Rdcase by perturbation of the Lebesgue measure, as in the proof below; for a general manifold, it holds if the injectivity radius of M is positive or, with additional technical issues, if the Ricci curvature of M is bounded below (see [8] or [17] and the references therein).
Corollary 4. Assume that there existc, α andδ >0 such that
∇U(x)·x≥ 1
c|x|α−c, |∇U(x)| ≤c(1∨ |x|δ) (9) for all x∈Rd. Then for allγ ≥0, γ >1−α and for V(x) = (1 +|x|2)−β/2eU(x)/2 withβ ∈R there exist C and s0>0 such that
Z
f2dµ≤s Z
|∇f|2(1∨ |x|2γ)dµ+ C sp
Z
|f|V dµ2
(10) for all smooth f and s≤s0. Here
p= 2β∨0 +d[δ∨(α+γ−1)]
2(α+γ−1) ·
The first hypothesis in (9) allows the computation of an explicit Lyapunov function W as in Definition 1, with an explicit mapφ, hence ψ in (8); the second hypothesis is made here only to obtain an explicit maphin (8), and then thes−pdependence as in the super Poincar´e inequality (10). Observe also from the proof that s0 does not depend on β, and that the constantCobtained by tracking in the proof its dependence on the diverse parameters would certainly be far from being optimal, as it is always the case in Lyapunov condition arguments.
A variant of the argument leads to a (weighted if α < 1) Poincar´e inequality for the measure µ, see Remark 13 below. For α >1 we can take γ = 0 in Corollary 4, obtaining a super-Poincar´e inequality with the usual Dirichlet form and a weightV, and then non uniform off-diagonal bounds on the Markov kernel density of the associated semigroup:
Theorem 5. Assume that∆U− |∇U|2/2 is bounded from above, and that there existc, δ >0 andα >1 such that
∇U(x)·x≥ 1
c|x|α−c, |∇U(x)| ≤c(1∨ |x|δ)
for allx∈Rd. Then for allt >0the Markov kernel of the semigroup(Pt)t≥0 admits a density pt(x, y)which satisfies the following bound : for all β∈Randε >0 there exists a constantC such that
pt(x, y)≤ C tp
eU(x)/2eU(y)/2
(1 +|x|2)β/2(1 +|y|2)β/2e−|x−y|
2 4(1+ε)t
for all0< t≤1 and almost every(x, y) in Rd×Rd. Here p= 2β∨0 +d[δ∨(α−1)]
2(α−1) ·
In particular, ifδ ≤α−1 (as in Example 8 below), and forβ = 0, we obtain p=d/2 as in the ultracontractive case of the heat semigroup. The bound on the kernel density derived in Theorem 5 will be proved for all t >0, but with an extra eCt factor, hence is relevant only for small times. For larger times it can be completed as follows :
Remark 6. In the assumptions and notation of Theorem 5, the measureµsatisfies a Poincar´e inequality; there exists a constant K >0 and for allt0 >0 there exists C=C(t0) such that for all t≥t0 and almost every (x, y) inRd×Rd
|pt(x, y)−1| ≤Ce−Kt eU(x)/2eU(y)/2 (1 +|x|2)β/2(1 +|y|2)β/2·
Remark 7. Our method can be extended from the case of diffusion semigroups to more general cases. For instance the weighted Nash inequalities can be derived for discrete val- ued reversible pure jump process as then a local super-Poincar´e inequality can easily be obtained. Then we should suppose that the Lyapunov condition holds forξ(w) =wand use [8, Lem. 2.12] instead of [7, Lem. 2.10] as in the proof below. Observe moreover that [6] has shown how to extend Davies’ method [12] for off-diagonal bounds to non diffusive semigroups.
Example 8. The measures with density Z−1e−u(x)α for α > 0 and u C1 convex such that Re−udx <+∞ satisfy the first hypothesis in (9) in Corollary 4 and Theorem 5. Indeed, by [1, Lem. 2.2],
∇u(x)·x≥u(x)−u(0)≥K|x| (11) withK >0 and for|x|large, so
∇(uα)(x)·x|x|−α =α u(x)α−1∇u(x)·x|x|−α≥αu(x)
|x|
α
1− u(0) u(x)
≥αK 2
α1 2 >0 for |x|large.
In particular for u(x) = (1 +|x|2)1/2, the hypotheses of Theorem 5 hold for α > 1 and δ =α−1 (andU =uα has a curvature bounded by below), thus recovering the on-diagonal bounds given in [18] (and [2] in dimension 1), and further giving the corresponding off-diagonal estimates. It was observed in [2] that in the limit case α = 1 the spectrum of −L does not only have a discrete part, so an on-diagonal bound such as pt(x, x) ≤ C(t)V(x)2 with V in L2(µ) can not hold. Forα >2 the semigroup is known to be ultracontractive (see [14] or [3, Sect. 7.7] for instance), and adapting our method to this simpler case were V = 1 leads to the corresponding off-diagonal bounds, see Remark 12.
For Cauchy type distributions, Proposition 2 specifies as follows : Corollary 9. Assume that there existα, c andδ >0 such that
lim inf
|x|→+∞∇U(x)·x≥d+α and |∇U(x)| ≤c(1∨ |x|δ), x∈Rd.
Then for all γ > 2/α and for V(x) = (1 +|x|2)−β/2eU(x)/2 with β > 0 there exist C and s0 >0 such that
Z
f2dµ≤s Z
|∇f|2(1∨ |x|2+2/γ)dµ+ C sp
Z
|f|V dµ2
for all smooth f and s≤s0. Here p=γ β+d[1∨(δ γ)]/2.
A variant of the argument also gives a weighted Poincar´e inequality for the measureµ, see Remark 13.
Example 10. The measures with density Z−1u(x)−(d+α) for α > 0 and u C2 convex such that R
e−udx <+∞ satisfy the first hypothesis in Corollary 9. Indeed, by (11),
∇U(x)·x= (d+α)∇u(x)
u(x) ·x≥(d+α)u(x)−u(0)
u(x) = (d+α)
1− u(0) u(x)
≥(d+α)(1−ε) for anyε >0, and for |x|large. In particular, choosing u(x) = (1 +|x|2)1/2, the hypotheses of Corollary 9 hold with δ = 0 for the generalized Cauchy measures with density Z−1 (1 +
|x|2)−(d+α)/2, where α >0.
The rest of this note is devoted to the proofs of these statements and further remarks.
Notation. If ν is a Borel measure, p ≥1 and r > 0 we let k · kp,ν be the Lp(ν) norm and Z
r
f dν = Z
|x|≤r
f(x)dν(x).
Proof of Proposition 2. We adapt the strategy of [8, Th. 2.8], writing Z
f2dµ= Z
r
f2dµ+ Z
|x|>r
f2dµ
forr >0. By assumption on φand W, and letting Φ(r) = inf|x|≥rφ(x), the latter term is Z
|x|>r
f2φ 1
φdµ ≤ 1 Φ(r)
Z
|x|>r
f2φ dµ≤ 1 Φ(r)
Z
f2φ dµ
≤ 1
Φ(r)
Z −LW
ξ(W)f2dµ+b Z
r0
f2dµ
≤ 1 Φ(r)
Z
|∇f|2 1
ξ′(W)dµ+b Z
r
f2dµ
by integration by parts (as in [7, Lem. 2.10]) and for r≥r0. Hence for all r≥r0
Z
f2dµ≤ 1 + b
Φ(r) Z
r
f2dµ+ 1 Φ(r)
Z
|∇f|2ω dµ (12) for ω= 1∨1/ξ′(W).
Now for all r > 0 the Lebesgue measure on the centered ball of radius r satisfies the following super-Poincar´e inequality :
Z
r
g2dx≤u Z
r
|∇g|2dx+b(r, u) Z
r
|g|dx 2
(13) for all u >0 and smoothg, where b(r, u) =cd(u−d/2+r−d) for a constantcd depending only on d. Forr= 1 (say) this is a linearization of the (Gagliardo-Nirenberg-) Nash inequality
Z
1
g2dx1+2/d
≤cd Z
1
(g2+|∇g|2)dx Z
1
|g|dx4/d
for the Lebesgue measure on the unit ball; then the bound and the value of b(r, u) for any radius r follow by homogeneity.
Hence, forg=f e−U/2, Z
r
f2dµ = Z
r
(f e−U/2)2dx
≤ 2u Z
r
|∇f|2e−Udx+u 2
Z
r
f2|∇U|2e−Udx+b(r, u) Z
r
eU/2
V |f|V e−Udx
!2
≤ 2u Z
|∇f|2ω dµ+u 2 h(r)
Z
f2dµ+b(r, u)g(r) Z
|f|V dµ 2
. By (12) and collecting all terms, it follows that for all r≥r0 and u >0
1−u
2h(r) 1 + b
Φ(r) Z
f2dµ ≤ 1
Φ(r) + 2u 1 + b
Φ(r) Z
|∇f|2ω dµ +
1 + b Φ(r)
b(r, u)g(r) Z
|f|V dµ 2
. (14) Now, forr≥r0, then Φ(r)≥Φ(r0) so 1+b/Φ(r)≤1+b/Φ(r0) := 1/k. Hence, ifu h(r)≤k, then the coefficient on the left hand side in (14) is greater than 1/2; on the other hand, is 2uΦ(r)≤k, then the coefficient of the energy is smaller than 2/Φ(r).
Let nows≤s0:= 4/Φ(r0) be given. Choosingr :=ψ(4/s) and thenu:=k min{1/h(r), s/8}, we observe thatr ≥r0 and u≤ks0/8, so
b(r, u) =cd(u−d/2+r−d)≤cd(u−d/2+r0−d)≤Cu−d/2
for a constant C depending only on d and r0. Hence, for these r and u, (14) ensures the existence of constants C ands0 >0 such that
Z
f2dµ≤s Z
|∇f|2ω dµ+Cg(r) min{1/h(r), s}−d/2 Z
|f|V dµ 2
for all weight V,f ands≤s0. This concludes the proof of Proposition 2.
Proof of Corollary 4. LetW ≥1 be aC2 map onRdsuch thatW(x) =ea|x|α for|x|large, where a >0 is to be fixed later on. Then
LW(x) =aα|x|2α−2W(x)
(d+α−2)|x|−α+aα− |x|−α∇U(x)·x for |x|large, by direct computation. Now
aα− |x|−α∇U(x)·x≤aα− 1 2c <0
for |x| ≥(2c2)1/α by the first assumption in (9), and for a <(2cα)−1, so for such ana there exists a constantC >0 for which
LW(x)≤ −C|x|2α−2W(x) (15)
for|x|large.
In other words, ifα >1, then the Lyapunov condition of Definition 1 holds withξ(u) =u andφ(x) =C|x|2α−2: observe indeed thatφ(x)→+∞ as|x| →+∞ forα >1.
If the general case when possiblyα≤1, then we letγ ≥0 andξbe aC1positive increasing map on (0,+∞) with ξ(u) =u(logu)−2γ/α for, say, u > e4γ/α. Then, by (15),
LW
ξ(W) = LW
a−2γ/α|x|−2γW ≤ −C|x|2(α+γ−1)
for |x| large, so the Lyapunov condition of Definition 1 holds with φ(x) = C|x|2(α+γ−1) if moreover γ >1−α.
Hence Proposition 2 ensures the super-Poincar´e inequality (8) forµ, with the weightω(x) = 1∨1/ξ′(W(x))≤C(1∨|x|2γ). We now chooseV(x) = (1+|x|2)−β/2eU(x)/2 withβ ∈R. Then, in the notation of Proposition 2, ψ(r) =Cr1/2(α+γ−1) for allr,g(r) = (1 +r2)β ≤2βr2β for r ≥ 1 if β >0, and g(r) ≤ 1 if β ≤0; finally h(r) ≤c2r2δ under the second assumption in
(9). This concludes the proof of Corollary 4.
Proof of Theorem 5. It combines and adapts ideas from [2] and [12], replacing the family of logarithmic Sobolev inequalities used in [12] by the super-Poincar´e inequalities (10). The positive constants C may differ from line to line, but will depend only onU and V.
The proof goes in the following several steps.
1. Let f be given in L2(µ). With no loss of generality we assume that f is non negative and C2 with compact support, and satisfies
Z
f V dµ = 1. Let also ρ > 0 be given and ψ be a C2 bounded map onRd such that |∇ψ| ≤1 and |∆ψ| ≤ ρ (the formal argument would consist in letting ψ(x) = x·n for a unit vector n in Rd). For a real number a we also let ϕ(x) =eaψ(x). We finally letF(t) =ϕ−1Pt(ϕf).
1.1. Evolution of R
F(t)V dµ. We first observe that d
dt Z
F(t)V dµ= Z
V ϕ−1LPt(ϕf)dµ= Z
L(V ϕ−1)Pt(ϕf)dµ (16) by integration by parts. Indeed, following the proof of [2, Cor. 3.1], two integrations by parts on the centered ball Br with radius r >0 ensure that
Z
Br
V ϕ−1LPt(ϕf)dµ= Z
Br
L(V ϕ−1)Pt(ϕf)dµ +
Z
Sd−1
hPt(ϕf)(rω)∇(V ϕ−1)(rω)·n−(V ϕ−1)(rω)∇Pt(ϕf)(rω)·ni
e−U(rω)rd−1dω for the inward unit normal vectorn. Then a lower boundλ∈Ron the Hessian matrix of U yields the commutation property and bound
|∇Pt(ϕf)| ≤e−λtPt|∇(ϕf)| ≤Ce−λt
by our assumption onf and ϕ; moreover, on the sphere|x|=r, both (V ϕ−1)(x)e−U(x)|x|d−1 and |∇(V ϕ−1)(x)|e−U(x)|x|d−1 are uniformly bounded byC(1 +rC) sup|x|=re−U(x)/2.In turn this term goes to 0 as r goes to infinity since for instance
U(x)−U(0) =
Z r−1(2c2)1/α
0
∇U(tx)·x dt+ Z 1
r−1(2c2)1/α
∇U(tx)·tx t−1dt≥ rα C −C for all |x|=r large enough, by assumption on∇U. Hence both boundary terms go to 0 asr goes to infinity; this proves (16).
Then we observe that
L(V ϕ−1)
V ϕ−1 =L(log(V ϕ−1)) +|∇log(V ϕ−1)|2. But
log(V ϕ−1) = U(x)
2 −βlog<x>−aψ(x) where<x>= (1 +|x|2)1/2, so by direct computation
L(log(V ϕ−1)) +|∇log(V ϕ−1)|2
=1
2∆U−1
4|∇U|2−βd <x>−2+β(2 +β)<x>−4|x|2−a∆ψ−2aβ <x>−2∇ψ·x+a2|∇ψ|2
≤ C+ (ρ+|β|)|a|+a2:=K
for allx if ∆U− |∇U|2/2 is bounded from above, |∆ψ| ≤ρ and|∇ψ| ≤1. HenceL(V ϕ−1)≤ K V ϕ−1, so
d dt
Z
F(t)V dµ≤K Z
F(t)V dµ, which implies
Z
F(t)V dµ≤eKt Z
F(0)V dµ=eKt Z
f V dµ=eKt. 1.2. Evolution of y(t) :=R
F(t)2dµ. By integration by parts, y′(t) = 2
Z
F ϕ−1LPt(ϕf)dµ=−2 Z
∇(F ϕ−1)· ∇Pt(ϕf)dµ
= −2 Z
e−aψ(∇F−aF∇ψ)·eaψ(∇F+aF∇ψ)dµ
= −2 Z
|∇F|2dµ+ 2a2 Z
|∇ψ|2F2dµ≤ −2 Z
|∇F|2dµ+ 2a2y(t) since |∇ψ| ≤1.But, for α >1 we can takeγ = 0 in Corollary 4, so that
Z
|∇F|2dµ≥s−1hZ
F2dµ− c sp
Z
F V dµ2i for any s≤s0 and a constant c=c(U, V), and then
Z
|∇F|2dµ≥u y(t)−c up+1e2Kt for any u(=s−1)≥u0:=s−10 .Herep=
2β∨0 +d[δ∨(α−1)]
/2(α−1).Hence y′(t)≤ −2(u y(t)−c up+1e2Kt) + 2a2y(t)
for any u≥u0. As long as y(t)≥c(p+ 1)up0e2Kt then the bracket is minimal at u= y(t)
C(p+ 1) 1/p
e−2Kt/p≥u0, and for this u
y′(t)≤ −Ce−2Kt/py(t)1+1/p+ 2a2y(t).
Then the mapz(t) =e−2a2ty(t) satisfies
z′(t)≤ −Ce−ktz(t)1+1/p
where
k:= 2
p(K−a2) = 2
p(C+ (|β|+ρ)|a|))>0.
It follows by integration that
z(t)−1/p≥z(0)−1/p+C p
1−e−kt
k ≥ C
p
1−e−kt k so that
y(t) =e2a2tz(t)≤Ce2a2t k 1−e−kt
p
≤Ce2a2tekt t
p
= C tpe2Kt.
This last bound holds as long asy(t)≥c(p+ 1)up0e2Kt, and then for all tprovided we take a possibly larger constantC (still depending only onU andV) in it and in the definition ofK.
1.3. In other words, for such a functionf: Z
ϕ−1Pt(ϕf)2
dµ≤c(t)2 for all t >0, where
c(t)2 = C
tpe2Kt, K =C+ (ρ+|β|)|a|+a2.
2. Duality argument. Let ϕ be defined as in step 1. By homogeneity and the bound
|Pt(ϕf)| ≤Pt(ϕ|f|), it follows from step 1 that Z
ϕ−1Pt(ϕf)2
dµ≤c(t)2Z
|f|V dµ2
for allt >0 and all continuous functionf with compact support.
Let nowt >0 be fixed,Qdefined byQf =ϕ−1Pt(ϕf) and W =c(t)V. Then
kQfk2,µ≤ kf Wk1,µ. (17)
The bound also holds for−a, so forϕ−1 instead of ϕ, so for ϕPt(ϕ−1f) = Q∗f where Q∗ is the dual ofQinL2(µ):
kQ∗fk2,µ ≤ kf Wk1,µ. (18) Following [2, Prop. 2.1] we considerRf = W−1Q(W f), and its dual in L2(W2µ), which is R∗f =W−1Q∗(W f). Then it follows from (18) that
kR∗fk2,W2µ≤ kfk1,W2µ
and then by duality that
kRfk∞,W2µ≤ kfk2,W2µ. Moreover (17) ensures that
kRfk2,W2µ≤ kfk1,W2µ, so finally
kR2fk∞,W2µ≤ kRfk2,W2µ≤ kfk1,W2µ.
As a consequence, by [3, Prop. 1.2.4] for instance, there exists a kernel densityr2(x, y) on Rd×Rd such that r2(x, y)≤1 forW2µ⊗W2µ-almost every (x, y) in Rd×Rdand
R2f(x) = Z
f(y)r2(x, y)W(y)2dµ(y)
for allf and W2µ-almost every x, hence (Lebesgue) almost every x since W ≥1 and µhas positive density.
Observing thatPtf =ϕ Q(ϕ−1f) and Q(g) =W R(W−1g), it follows that P2tf(x) = ϕ(x)Q2(ϕ−1f)(x) =ϕ(x)W(x)R2(W−1ϕ−1f)(x)
= ϕ(x)W(x) Z
ϕ(y)−1W(y)f(y)r2(x, y)dµ(y)
for allf and almost every x. Hence the Markov kernel ofP2t has a density with respect toµ, given by
p2t(x, y) =ϕ(x)ϕ(y)−1W(x)W(y)r2(x, y)≤ea(ψ(x)−ψ(y))c(t)2V(x)V(y).
3. Conclusion. It follows from step 2 that the semigroup (Pt)t≥0 at timet >0 admits a Markov kernel density pt(x, y) with respect to µ, such that for all real numbera and all C2 bounded mapψon Rd with|∇ψ| ≤1 and|∆ψ| ≤ρ, the bound
pt(x, y)≤ C
tpV(x)V(y)eKt+a(ψ(x)−ψ(y)) (19)
hold for almost every (x, y) inRd×Rd, withK =C+ (ρ+|β|)|a|+a2.
We now lett >0,x and y be fixed withy 6=x. Letting r =|x| ∨ |y|and n= |y−x|y−x, we let ψbe aC2 bounded map onRdsuch that |∇ψ| ≤1 and |∆ψ| ≤ρ everywhere, and such that ψ(z) =z·n if|z| ≤r. For instance we leth(z) be a C2 map onRwith h(z) =z if|z| ≤r,h constant for|z| ≥Rand satisfying|h′| ≤1 and|h′′| ≤ρ, which is possible forRlarge enough compared to ρ−1; then we let ψ(z) =h(z·n).
Such a mapψsatisfiesψ(x)−ψ(y) =−|x−y|, so the quantity in the exponential in (19) is
−a|x−y|+ (a2+ (ρ+|β|)|a|)t+Ct.
Since ρ > 0 is arbitrary (and the constant C depends only on U and V) we can let ρ tend to 0. Then we use the bound
|aβ| ≤εa2+ 1 4εβ2
and optimise the obtained quantity by choosinga=|x−y|/(2t(1 +ε)), leading to the bound pt(x, y)≤ C
tpV(x)V(y) exph
− |x−y|2
4(1 +εt) +β2t 4ε +Cti
and concluding the argument.
Remark 11. The computation in steps 1 and 2 of this proof could be written in the more general setting of a reversible diffusion generator L on a space E with carr´e du champ Γ, under the assumptions Γ(ψ)≤1, L(V ϕ−1)≤KV ϕ−1 and
Z
g2dµ≤s Z
Γ(g)dµ+Cs−pZ
|g|V dµ2
for all g and s≤s0. Then step 3 would yield the corresponding bound with|x−y|replaced by the intrinsic distance
ρ(x, y) = sup
|ψ(x)−ψ(y)|; Γ(ψ)≤1 .
Proof of Remark 6. First observe that, under the first hypothesis in (9) in Corollary 4, withα≥1, then (15) in the proof of this corollary ensures that
LW
W (x)≤ −C+b1|x|≤r0
for all x and for positive constants b and C. This is a sufficient Lyapunov condition forµto satisfy a Poincar´e inequality (see [1, Th. 1.4]).
Then we can adapt an argument in [3, Sect. 7.4], that we recall for convenience. We slightly modify the notation of the proof of Corollary 4, letting a = 0 (hence ϕ = 1), and for anyt≥0, letting Rtf =V−1Pt(V f) and R∞f =V−1
Z
V f dµ. Then, by the Poincar´e inequality forµ, there exists a constant K >0 such that
kRtf −R∞fk22,V2µ= Z
Pt(V f)− Z
V f dµ
2dµ ≤ e−2Kt Z
V f −
Z
V f dµ
2dµ
≤ e−2Kt Z
|V f|2dµ=e−2Ktkfk22,V2µ
for allt≥0. Moreover
kRt0fk2,V2µ≤c(t0)kfk1,V2µ, kRt0fk∞,V2µ≤c(t0)kfk2,V2µ for allt0 >0, by step 2 in the proof of Corollary 4, so
kRt+2t0f −R∞fk∞,W2µ=kRt0RtRt0f −Rt0R∞Rt0fk∞,W2µ
≤c(t0)kRtRt0f −R∞Rt0fk2,W2µ≤c(t0)e−KtkRt0fk2,W2µ≤c(t0)2e−Ktkfk1,W2µ
for all t ≥ 0 and t0 > 0. Changing t+ 2t0 into t ≥ t0 and writing the kernel density of Rt−R∞ in terms of the kernel density pt(x, y) of Pt lead to the announced bound on the
densitypt(x, y)−1.
Remark 12. The proof of Theorem 5 simplifies in ultracontractive situations where one takes 1 as a weightV. For instance, for the heat semigroup on Rd, one starts from the (non weighted) super Poincar´e inequality for the Lebesgue measure onRd:
Z
F2dx≤u Z
|∇F|2dx+cdu−d/2Z
|F|dx2
for all u >0 and for a constant cd depending only on d(a linearization of the Nash inequal- ity (1) for the Lebesgue measure on Rd, which can also be recovered by letting r go to +∞
in (13)). The constantK obtained in step 1.1 is K=ρ|a|+a2, and the very same argument leads to the (optimal int) off-diagonal bound
pt(x, y)≤C t−d/2e−|x−y|
2 4t
for allt >0 (also derived in [3, Sect. 7.2] for instance with the optimal C = (4π)−d/2 when starting from by the Euclidean logarithmic Sobolev inequality).
The argument can also be written in the ultracontractive case of U(x) = (1 +|x|2)α/2 with α >2 : one starts from the super Poincar´e inequality
Z
F2dµ≤u Z
|∇F|2dµ+ec 1+u−
2α−2α Z
|F|dµ2
for allu >0 (see [17, Cor. 2.5] for instance). Then one obtains the off-diagonal bound pt(x, y)≤eC 1+t−
α−2α e−|x−y|
2 4t
for allt >0, also derived in [3, Sect. 7.3] by means of an adapted entropy-energy inequality.
Proof of Corollary 9. Following the strategy of the proof of Corollary 4, we observe that LW(x) ≤ −C|x|a−2 with C > 0 for |x| large, if W(x) = |x|a with a < 2 +α. Then, for ξ(r) =r1−b,LW/ξ(W)≤ −C|x|ab−2, and we are in the framework of Proposition 2 provided ab >2, a <2 +α and 0≤b <1. We finally letγ = 2/(ab−2).
Remark 13. We have seen in Remark 6 that, by (15), the sole first hypothesis in (9) in Theorem5, forα≥1, ensures a Poincar´e inequality for µ.
In the case when 0< α <1, let us takeγ = 1−αand replace ω(x) =C(1∨ |x|2γ) by aC1 map ω(x), still equal to C|x|2γ for|x|large, and let
Lωf =ω∆f −(ω∇U− ∇ω)· ∇f =ωLf+∇ω· ∇f
be the generator with symmetric invariant measure µ. Then, still under the first hypothesis in (9), (15) implies that
LωW
W =ωLW
W +∇ω· ∇logW ≤ −C+c|x|−α≤ −C′ (20) for |x|large and C′ >0. This now leads to a Poincar´e inequality forµ and for the energy
− Z
f Lωf dµ= Z
|∇f|2ω dµ≤C Z
|∇f|2(1 +|x|2(1−α))dµ,
hence deriving, under the sole first hypothesis in (9), the weighted Poincar´e inequality ob- tained in [7, Prop. 3.6] for measures as in Example 8.
Likewise, under the sole first hypothesis in Corollary 9, the weight ω(x) = 1 +|x|2 satisfies the conclusion of (20), leading to a Poincar´e inequality for µand for the energy
− Z
f Lωf dµ=C Z
|∇f|2(1 +|x|2)dµ, that is, to the weighted Poincar´e inequality
Z f −
Z
f dµ2
dµ≤C Z
|∇f|2(1 +|x|2)dµ
for measures as in Example 10, withα >0. For measuresµwith densityZ−1(1+|x|2)−(d+α)/2, this inequality has been derived in [7, Prop. 3.2] for anyα >0. The dependence indand αof the constants C obtained here and in [7] is not tractable. An explicit constant C =C(d, α) has been derived in [4, Th. 3.1] for α ≥d, and improved in a sharp manner in [15, Cor. 14]
for α≥d+ 2.
Acknowledgments. The authors warmly thank I. Gentil for enlightening discussion on this and related questions.
They are grateful to the two referees for a careful reading of the manuscript and most relevant comments and questions which helped improve the presentation of the paper. This note was partly written while they were visiting the Chinese Academy of Sciences in Beijing; it is a pleasure for them to thank this institution for its kind hospitality. The first two authors also acknowledge support from the French ANR-12-BS01-0019 STAB project.
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Ceremade, Umr Cnrs 7534, Universit´e Paris-Dauphine, Place du Mar´echal de Lattre de Tas- signy, F-75775 Paris cedex 16
E-mail address: bolley@ceremade.dauphine.fr
Institut Universitaire de France and Laboratoire de Math´ematiques, Umr Cnrs 6620, Uni- versit´e Blaise Pascal, Avenue des Landais, F-63177 Aubi`ere cedex
E-mail address: guillin@math.univ-bpclermont.fr
Huazhong University of Science and Technology, Wuhan.
E-mail address: wang@math.univ-bpclermont.fr
